Econ 57 Spring 1998 Midterm

1. Without doing any calculations, explain why a pie chart and a bar chart would both be poor choices for displaying these data on the number of U.S. commercial banks of different sizes.

Assets (Dollars) Number of Banks
0 to 24.9 million 3,330    
25 to 49.9 million 3,145    
50 to 99.9 million 2,782    
100 to 499.9 million 2,461    
500 to 999.9 million 253    
1 to 2.9 billion 202    
3 to 9.9 billion 172    

2. Carefully explain the error in this statistical argument by Henry Van der Eb, president of the Mathers Fund, for why stock prices are due to fall: "We've been spoiled by huge stock returns; in order to return to the [long-term] mean of 9.7% returns, we must have some down years." [quoted in Craig Torres, "Three Cash-Laden Bears See End of Bulls' Run," The Wall Street Journal, August 4, 1992.]

3. Answer this job-interview question that was asked this year: "You are playing tennis against someone who is not as good as you. Assuming that the games are independent and that your probability of winning any game is constant and greater than 0.5, are you more likely to win more games than your opponent if you play 5 games or if you play 10 games?" Explain your reasoning.

4. Here is a job-interview question that was asked this year: "There is a 60% chance of rain on Saturday and a 40% chance of rain on Sunday. What is the probability that it will not rain this weekend." The interviewer said that the correct answer is (1 - 0.6)(1 - 0.4) = 0.24. The job candidate said that the correct answer is 1 - 0.6(0.4) = 0.76. What do you say?

5. Answer this job-interview question that was asked last year: "A box contains 20 pennies, of which 19 are normal coins with a head on one side and tail on the other; one coin has heads on both sides. The box is shaken until the coins are mixed thoroughly. Then one coin is randomly selected and flipped five times. Each time it lands heads. What is the probability that it is the two-headed coin?"

6. Answer this centuries-old question that the Chevalier de Mere asked Blaise Pascal: Two-evenly matched players are playing a sequence of games. The first person to win 4 games wins $1000. They have played 4 games and A has won 3 games and B has won 1 game. At this point they are forced to stop playing. How should they divide the $1000, so that each person receives the expected value of what they would have won had they continued playing?

7. An article in the San Francisco Chronicle [September 6, 1984] suggested that AIDS may be caused by drinking fluoridated water: "While half the country's communities have fluoridated water supplies and half do not, 90 percent of AIDS cases are coming from fluoridated areas and only 10 percent are coming from nonfluoridated areas." Why are these data unpersuasive? Be specific.

8. A school estimated the average number of school-age children per family having school-age children by questioning each child in the school and calculating the mean of their answers. Was their estimate too high or too low? Use a simple numerical example to explain your reasoning.

9. The random walk hypothesis says that the probability that stock prices will increase today is independent of the behavior of stock prices on other days. If, on any given day, there is a 0.52 probability that the Dow Jones Industrial Average will go up and a 0.48 probability that it will go down (and no chance that it will be unchanged), what is the probability that during the course of a year with 250 trading days the Dow will have more up days than down days? Use a normal approximation.

10. Identify several problems that prevent this graphic from conveying useful information [The Student Life December 11, 1996.]: